In the episode where Miley loses her voice, the doctor tells her if she doesn't get surgery she can never sing again. But how did he know Miley is a singer?
There are people, (grown ups) who know who she is. The dr. is making a house call which doesn't happen anymore. I think it is believable that the dr. knows who she is and comes to the home (paid well) because he knows how serious the situation could be. Kind of a special arrangement between the dr. and the Stewarts. Otherwise, why not go to the dr. office just like everyone else.